A Power-Law Diffusion Between Arithmetic and Geometric Brownian Motion

Summary

The document introduces a diffusion model that can interpolate between arithmetic Brownian motion and geometric Brownian motion. In the proposed stochastic differential equation, the diffusion term is proportional to the state raised to a power γ. Setting γ to zero gives state-independent Brownian noise, while setting it to one makes volatility proportional to the state, as in geometric Brownian motion. Intermediate exponents produce a state-dependent diffusion between those endpoint cases.

The discussion is a short identification of a familiar model form in response to a question about derivative pricing. It does not specify the drift, assumptions about the state space, parameter estimation, derivative valuation formulas, or empirical evidence. The answer therefore clarifies the relationship between two diffusion models, but does not establish that this specification is suitable for any particular derivative or that it is a proprietary trading model.

Key ideas

• A diffusion with volatility proportional to the state raised to γ connects arithmetic and geometric Brownian motion.
• At γ equal to zero, the diffusion term is state independent.
• At γ equal to one, volatility scales with the state as in geometric Brownian motion.
• Intermediate γ values create intermediate forms of state dependence, but the document gives no pricing or calibration procedure.

Tags

# Mix of Arithmetic and Geometric Brownian Motion


# Mix of Arithmetic and Geometric Brownian Motion












Talking with some traders the other day, I found out that they were using a pricing model based on a mix between a geometric brownian motion and an arithmetic brownian motion to price certain derivatives.

Would anyone know if this is actually a well known model? Or is this actually one of these "secret" solutions that certain quant teams develop?

Practitioners' take on the matter would be particularly helpful.

## Answer by Kiwiakos (score 0, accepted)

https://quant.stackexchange.com/a/28240

Are you talking about something like this?

$$dx(t)=\ldots\ dt+[x(t)]^\gamma\ dW(t)$$

If $\gamma$ is zero then you've got BM, if it's one you get GBM, inbetween you have a 'mix'.

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